Abstract
Modern highresolution microscopes are commonly used to study specimens that have dense and aperiodic spatial structure. Extracting meaningful information from images obtained from such microscopes remains a formidable challenge. Fourier analysis is commonly used to analyze the structure of such images. However, the Fourier transform fundamentally suffers from severe phase noise when applied to aperiodic images. Here, we report the development of an algorithm based on nonconvex optimization that directly uncovers the fundamental motifs present in a realspace image. Apart from being quantitatively superior to traditional Fourier analysis, we show that this algorithm also uncovers phase sensitive information about the underlying motif structure. We demonstrate its usefulness by studying scanning tunneling microscopy images of a Codoped iron arsenide superconductor and prove that the application of the algorithm allows for the complete recovery of quasiparticle interference in this material.
Introduction
The past few decades have seen dramatic advances in the understanding of the structure of materials via scattering and microscopy techniques. Scattering techniques are useful when perfect periodicity exists in a material, while microscopy is well suited for specimens that lack periodicity. Recent advances in microscopy techniques, when coupled with improved computing power, have enabled the scientific community to generate massive, multidimensional spatial images of specimens as a function of control parameters, such as time, energy, and applied stimulus. Examples of such advanced tools include superresolution optical microscopy to inspect the structure of proteins beyond the diffraction limit^{1}, scanning transmission electron microscopy to examine the chemical structure of materials at the atomic scale^{2}, and scanning tunneling microscopy (STM) to visualize the quantum electronic structure of surfaces with atomic resolution^{3}. Fundamentally, a microscope image represents the interaction between the probe and the specimen, and often times sophisticated analysis must be performed to uncover the scientific content present in the image. Specimens of interest for STM studies include metals^{4}, twodimensional (2D) materials^{5}, unconventional superconductors^{6}, topological materials^{7,8}, and charge^{9} and spin^{10,11} ordered materials, among others. Image analysis of these materials has provided several unique insights into the quantum electronic structure and interactions present within them. Many microscopy techniques utilize the Fourier transform (FT)^{12,13} for analysis, revealing the characteristic wavelengths present in the image, which are then related to a scientific theory of the specimen being studied. When perfect periodicity exists in an image, the FT provides a concise and accurate description of the image^{6,7,9,14,14,15,16,17,18}. However, when applied to aperiodic images, the FT suffers from phase noise leading to a fundamental loss of information^{19,20}. With the proliferation of new computing techniques, one may wonder if the maturation of optimization algorithms can be leveraged to extract more information from a microscopy image than through the FT.
In this work, we consider a class of images that are of particular importance to microscopy—those that can be perceived as a basic motif, called a kernel, that is repeated aperiodically across the image. Examples of kernels include electronic scattering patterns around atomic defects (in STM) and fluorescence from individual proteins (in optical microscopy). We present the development of an algorithm, based on nonconvex optimization, for analyzing such images that quantitatively extracts the principal motifs present in an image. We demonstrate that this algorithm can elucidate fundamentally new information unavailable through traditional FT analysis. While our methods are generally applicable to a wide range of microscopy techniques, in this work we focus on its application to STM.
Results
STM and scanning tunneling spectroscopy
STM and scanning tunneling spectroscopy produces 2D spectroscopic maps of the local density of states (LDoS) at position x with energy ω, forming a threedimensional dataset. The contrast in these images stems from local spatial variations of the LDoS, denoted as δρ(x, ω). Measurements in which δρ(x, ω) is ascribed to material defects that cause electron scattering and interference^{4} are particularly interesting, and such maps are often called quasiparticle interference (QPI) maps. Analysis of QPI maps has uncovered information on the dispersion relations and scattering processes in semimetals^{21,22}, hightemperature superconductors^{6,14,18,23}, and other systems. Let us suppose that the LDoS pattern created by a single defect located at x with energy ω is δρ_{0}(x, ω), and that the STM image is composed from N defects located at x_{1}, x_{2}, …, x_{N}:
where c_{j} are constants. A realworld example of such an image is shown in Fig. 1a, obtained on the pnictide superconductor NaFeAs^{18}.
According to scattering theory, the FT of the QPI image of an individual defect, δρ_{0}(q, ω) = ∫dx e^{−iq⋅x}δρ_{0}(x, ω), is correlated with the underlying electronic structure of the material^{14,15,16}. Many quantum materials previously studied by STM, examples of which include superconducting cuprates^{6}, pnictides^{18} and chalcogenides^{10}, charge density wave materials^{9}, topological insulators^{7}, and correlated oxides^{24} have sufficient disorder so that the LDoS signatures of different defects overlap. In this situation, it is not possible to identify the isolated defect signature through inspection. Instead, the traditional analysis^{9} proceeds by taking the FT of the entire STM image δρ(x, ω) in (1):
While the quantity of interest for QPI analysis is δρ_{0}(q, ω), the experimental FT image contains a frequencyvarying, complexvalued phase factor, \({\mathcal{P}}({\bf{q}})\equiv {\sum }_{j=1}^{N}{c}_{j}\exp \{i{\bf{q}}\cdot {{\bf{x}}}_{j}\}\). This is illustrated in Fig. 1b, where the real part of the FT (ReFT) displays wild oscillations due to \({\mathcal{P}}({\bf{q}})\). To mitigate this, the magnitude of the FT (magFT) is taken, and the analysis proceeds by assuming that \({\mathcal{P}}({\bf{q}})\) is approximately constant in magnitude so that \(\left{\mathcal{P}}({\bf{q}})\right\approx \bar{c}\sqrt{N}\), where \(\bar{c}\) is the average value of the c_{j}. The result of this procedure is illustrated in Fig. 1c, showing that debilitating noise still persists in the FT after taking the modulus. Moreover, the procedure of obtaining the magFT effectively eliminates half of the useful information in the complexvalued FT, annihilating all of the phase information from electron scattering processes originally present in real space. Intense peaks and contours in the real and imaginary parts of δρ(q, ω) are experimental indicators of dominant scattering wavevectors and order parameter symmetries, which can reveal important properties about the superconducting gap function sign structure^{25,26} and surface states of topological insulators^{27,28}. However, random phase noise fluctuations in experimental QPI spectra make comparisons with theoretical QPI calculations difficult^{29}.
An improved analysis technique to FTSTM would identify the location and the LDoS signature associated with each defect in a quantitatively rigorous fashion that respects experimental and materialspecific constraints. For instance, defects remain fixed in position across a series of STM images in which the measurement bias voltage is varied (see Fig. 2). In this work, we present an analysis technique based on nonconvex optimization that possesses these desirable features while being broadly applicable to other forms of microscopy and image analysis.
Connection with sparse blind deconvolution
Our algorithm is based on a deconvolutional procedure illustrated in Fig. 2. The image (denoted as \({\mathcal{Y}}\)) in Fig. 2a was produced by simulating the effect of quasiparticle scattering from numerous point defects randomly distributed across the image. At a given bias voltage, the image consists of a recurrent scattering pattern (called the kernel \({{\mathcal{A}}}_{0}\)) convolved with the locations and relative weights of each defect (called the activation map \({{\mathcal{X}}}_{0}\)) as illustrated in Fig. 2a and represented as \({\mathcal{Y}}={{\mathcal{A}}}_{0}\star {{\mathcal{X}}}_{0}\). The underlying challenge in our analysis is to invert the procedure—starting with an STM image, determine the kernel and its corresponding activation map. The kernel and activation map are easily identified by inspection in Fig. 2a; however, this becomes a highly nontrivial problem in the presence of many overlapping kernels and experimental noise. Similar convolutional models are used in neuroscience to model neuron spike patterns^{19} and in systems biology to capture responses of the endocrine system^{20}. In contrast, our algorithm focuses on a 2D signal.
When \({{\mathcal{A}}}_{0}\) contains multiple slices, with each slice corresponding to a different bias voltage, we mathematically express the proposed model for STM measurements by collecting the convolutions for each voltage slice using the notation
which is schematically depicted in Fig. 2b. The activation map \({{\mathcal{X}}}_{0}\) is shared globally across all measurement biases and \({\mathcal{Z}}\) is an additive noise tensor. The task of recovering both \({{\mathcal{A}}}_{0}\) and \({{\mathcal{X}}}_{0}\) given \({\mathcal{Y}}\) is known as the sparse blind deconvolution (SBD) problem.
Formulating the SBD problem
Over the past decade, a wealth of heuristics and applications for sparse signal recovery have been developed, often leading to efficient algorithms in theory as well as in practice^{30,31} (see Supplementary Note 2). We investigate the following heuristic for producing estimates \(\hat{{\mathcal{A}}}\) and \(\hat{{\mathcal{X}}}\) for \({{\mathcal{A}}}_{0}\) and \({{\mathcal{X}}}_{0}\), by posing an optimization problem based on (3):
which allows one to recover \(\hat{{\mathcal{X}}}=\arg \ {\min }_{{\mathcal{X}}}{\psi }_{\lambda }(\hat{{\mathcal{A}}},{\mathcal{X}})\).
This is similar to prem .vious formulations proposed for various SBD applications: the Frobenius norm term \({\left\Vert \cdot \right\Vert }_{\mathrm{F}}^{2}\) promotes data fidelity upon minimization \(({\mathcal{Y}}\simeq \hat{{\mathcal{A}}} \otimes \hat{{\mathcal{X}}})\), and a regularization term \(r(\hat{{\mathcal{X}}})\) is chosen, such as the ℓ_{1} norm, so that the minimization encourages \(\hat{{\mathcal{X}}}\) to be sparse, with λ ≥ 0 governing the tradeoff between the two objectives. However, most SBD applications focus on signal enhancement that uses the convolutional model as a rough guideline, leading to a weak notion of accurate estimation^{32}. In contrast, the convolutional model fits naturally into the STM setting, in which robust, consistent results are paramount for scientific investigation. These considerations prompt a number of choices that are not emphasized in previous heuristics, such as the domain of \({\mathcal{A}}\), form of \(r(\hat{{\mathcal{X}}})\), and refinement of the estimates.
In order to solve this optimization problem, we present the SBDSTM algorithm:
Algorithm 1 Complete SBDSTM procedure
Input:

Observation \({\mathcal{Y}}\in {{\mathbb{R}}}^{{n}_{1}\times {n}_{2}\times s}\), kernel size \(\left({m}_{1},{m}_{2}\right)\), initial λ_{0} ≥ 0, decay rate \(\alpha \in \left[0,1\right)\), and final λ_{end} ≥ 0.
Initial phase:

1.
Randomly initialize: \({{\mathcal{A}}}^{\left(0\right)}\in {\mathcal{S}}={{\mathbb{S}}}^{{m}_{1}\times {m}_{2}\times s}\).

2.
\({{\mathcal{A}}}_{* }^{\left(0\right)},{{\mathcal{X}}}_{* }^{\left(0\right)}\leftarrow\)ASolve\(({{\mathcal{A}}}^{(0)},{\lambda }_{0},{\mathcal{Y}})\).
Refinement phase:

1.
Lifting: Get \({{\mathcal{A}}}^{\left(1\right)}\in {S}^{\prime}={{\mathbb{S}}}^{{m}_{1}^{\prime}\times {m}_{2}^{\prime}\times s}\) by zeropadding the edges of \({{\mathcal{A}}}_{* }^{\left(0\right)}\) with a border of width \(\left\lfloor \frac{{m}_{i}}{2}\right\rfloor\).

2.
Set λ_{1} = λ_{0}.

3.
Continuation: Repeat for k = 1, 2, … until λ_{k} ≤ λ_{end},

(a)
\({{\mathcal{A}}}_{* }^{\left(k\right)},{{\mathcal{X}}}_{* }^{\left(k\right)}\leftarrow\)ASolve\(\left({{\mathcal{A}}}^{\left(k\right)},{\lambda }_{k},{\mathcal{Y}},{{\mathcal{X}}}_{* }^{\left(k1\right)}\right)\),

(b)
Centering:

i.
Find the size m_{1} × m_{2} submatrix of \({{\mathcal{A}}}_{* }^{\left(k\right)}\) that maximizes the Frobenius (square) norm across all m_{1} × m_{2} submatrices.

ii.
Get \({{\mathcal{A}}}^{\left(k+1\right)}\) by shifting \({{\mathcal{A}}}_{* }^{\left(k\right)}\) so that the chosen m_{1} × m_{2} restriction is in the center, removing and zeropadding entries as needed.

iii.
Normalize \({{\mathcal{A}}}^{\left(k+1\right)}\) so it lies in \({{\mathcal{S}}}^{\prime}\).

iv.
Shift \({{\mathcal{X}}}_{* }^{\left(k\right)}\) along the antiparallel vector to the shift of \({{\mathcal{A}}}_{* }^{\left(k\right)}\).

i.

(c)
Set λ_{k+1} = αλ_{k}.

(a)
Output:

1.
Extract \(\hat{{\mathcal{A}}}\in {\mathcal{S}}\) by extracting the restriction of the final \({{\mathcal{A}}}^{\left(k+1\right)}\) to the center m_{1} × m_{2} window.

2.
Find the corresponding activation map \(\hat{{\mathcal{X}}}\in {{\mathbb{R}}}^{{n}_{1}\times {n}_{2}}\) by solving \({\min }_{{\mathcal{X}}}{\psi }_{{\lambda }_{k}}(\hat{{\mathcal{A}}},{\mathcal{X}})\).
Function Asolve
Input:

Current kernel, \({{\mathcal{A}}}_{{\rm{in}}}\), current sparsity parameter, λ_{in}, the observation \({\mathcal{Y}}\), current activation map \({{\mathcal{X}}}_{{\rm{in}}}\) (Refinement Phase)
Minimization
if Initial Phase then

1.
Initialize \({\mathcal{X}}\) as a zero matrix of size (n_{1}, n_{2})

2.
\({{\mathcal{X}}}_{1}\leftarrow\) Minimize \({\mathcal{X}}\) for \({\psi }_{{\lambda }_{{\rm{in}}}}({{\mathcal{A}}}_{{\rm{in}}},{\mathcal{X}})\) using the FISTA algorithm^{33}.
else
\({{\mathcal{X}}}_{1}\leftarrow {{\mathcal{X}}}_{{\rm{in}}}\)
end if

1.
\({{\mathcal{A}}}_{{\rm{out}}}\leftarrow\) Minimize \({\mathcal{A}}\) for \({\psi }_{{\lambda }_{{\rm{in}}}}({\mathcal{A}},{{\mathcal{X}}}_{1})\) using the Riemannian TrustRegion Method (RTRM) over the sphere^{34}.

2.
\({{\mathcal{X}}}_{{\rm{out}}}\leftarrow\) Minimize \({\mathcal{X}}\) for \({\psi }_{{\lambda }_{{\rm{in}}}}({{\mathcal{A}}}_{{\rm{out}}},{\mathcal{X}})\) using FISTA.
Output:

\({{\mathcal{A}}}_{{\rm{out}}}\), \({{\mathcal{X}}}_{{\rm{out}}}\).
See Supplementary Notes 2 and 3 for further discussion on formulating and solving Eq. (4), and Supplementary Note 4 for how our approach to SBD can be applied to image deblurring by using an objective similar to Eq. (4).
The blind deconvolution approach
To demonstrate the strength of SBDSTM, consider the situation illustrated in Fig. 3. We generated a simulated observation \({\mathcal{Y}}\) using a ground truth scattering pattern \({{\mathcal{A}}}_{0}\) similar to Fig. 1d and a dense, randomly generated activation map \({{\mathcal{X}}}_{0}\) shown in Fig. 3b. Convolving the truth data with the activation map and adding significant white noise with variance η so that the signaltonoise ratio (SNR) is less than unity, with \({\rm{SNR}}\equiv \frac{{\mathrm{var}}({{\mathcal{A}}}_{0})}{\eta }\), we generate the image shown in Fig. 3c. With many overlapping kernels and substantial noise, it is a futile task to accurately identify the underlying kernel and activation map of the image through visual inspection. However, as shown in Fig. 3d, e, SBDSTM successfully recovers a kernel \(\hat{{\mathcal{A}}}\) and its associated activation map \(\hat{{\mathcal{X}}}\) that closely resemble the truth data. The results shown in Fig. 3 were obtained with a fixed λ = 0.1. The scaling of the activation map entries is due to the choice of λ, and the noise also introduces blurring in the activation map^{35}. Despite this, the overall features of the activation map and the recovered kernel are remarkably similar to the ground truth.
In Fourier space, the ReFT of \({\mathcal{Y}}\) is missing crucial features of the true ReFT spectrum and has noise fluctuations ≈100 times that of the true transform. However, the ReFT of the SBDSTM recovered kernel is consistent with the true ReFT in both its structure and amplitude.
In our implementation, SBDSTM yields an activation map shared across all bias voltages. This not only reveals the spatial distribution of the defect kernels but also naturally improves the accuracy of the SBDSTM recovered kernels at bias energies with noisy measurements. Consequently, SBDSTM returns more physically meaningful results when data from multiple biases are simultaneously analyzed than if each constantbias slice of \({\mathcal{Y}}\) were individually analyzed. SBDSTM results on a simulated noisy STM dataset with 41 bias voltages are found in Supplementary Note 5, demonstrating that SBDSTM is successful in optimizing the objective function with STM constraints in mind.
Performance characterization
Before invoking SBDSTM on experimental data, we must understand its limitations and domain of applicability. The complexity of the STM deconvolution problem varies depending on the SNR and the overlap tendency of nearby defects. A series of numerical experiments on simulated data were performed to investigate the effects of defect concentration θ—the probability that any entry of \({{\mathcal{X}}}_{0}\) is truly nonzero—and additive measurement noise on the expected success of SBDSTM. Simulated STM images are produced in a similar fashion as in Fig. 3, and the performance of SBDSTM was assessed as a function of four adjustable parameters—the image size n ≡ n_{1} × n_{2}, kernel size m ≡ m_{1} × m_{2}, kernel concentration θ, and SNR. Details of the data generation and simulation work are contained in Supplementary Notes 1 and 6. To assess the accuracy of kernel recovery in real space, we define the realspace recovery error metric as \(\epsilon ({\hat{{\mathcal{A}}}}_{\theta ,\eta },{{\mathcal{A}}}_{0})\equiv \frac{2}{\pi }\arccos \langle {\hat{{\mathcal{A}}}}_{\theta ,\eta },{{\mathcal{A}}}_{0}\rangle\), with \(\langle {\hat{{\mathcal{A}}}}_{\theta ,\eta },{{\mathcal{A}}}_{0}\rangle\) denoting the inner product of the vectorizations of \({\hat{{\mathcal{A}}}}_{\theta ,\eta }\) and \({{\mathcal{A}}}_{0}\), which are the recovered and truth kernels, respectively. Figure 4a depicts a normalized defect size \(\frac{m}{n}\) vs. concentration θ phase diagram to explore the interplay between \(\frac{m}{n}\) and θ on realspace algorithmic accuracy \(\epsilon ({\hat{{\mathcal{A}}}}_{\theta },{{\mathcal{A}}}_{0})\) in noisefree (η = 0) simulated measurements. We observe a phase transition in SBDSTM performance in the \(\frac{m}{n}\theta\) plane. The bottom left of the plot captures situations in which the defects have negligible probability of overlapping, facilitating the nearperfect deconvolution of the noisefree image. Increasing either \(\frac{m}{n}\) or θ introduces error in the kernel recovery due to increased overlapping between defects. Practically, \(\frac{m}{n}\) can be reduced by increasing the overall STM measurement area n in an attempt to perform deconvolutionbyinspection. However, at high defect concentrations θ or moderate noise levels, this strategy cannot guarantee success, while SBDSTM can still return reliable estimates.
Next, we briefly discuss the SBDSTM performance as a function of noise in the signal. Figure 4b shows the evolution of \(\epsilon ({\hat{{\mathcal{A}}}}_{\theta ,\eta },{{\mathcal{A}}}_{0})\) as a function of defect concentration θ for three values of SNR ranging from noise free to noise dominated (SNR = 0.792). At high noise levels, performance error fluctuations in the far left of the plot appear because of the statistically futile challenge of accurately identifying lowdensity, lowintensity motifs under high levels of noise. The error curves appear to converge into a narrow band when θ ≳ 0.01, demonstrating that the algorithm is robust to a wide range of SNRs for higher concentrations. By θ ≈ 0.2, the defect concentrations are sufficiently dense so that virtually all defect kernels are overlapping, causing SBDSTM to collapse and return unreliable estimates. These trends persist when the kernel size and SNR are further increased, as described in Supplementary Note 6. Altogether, our simulations show that in a wide range of parameters (Fig. 4a), SBDSTM performs splendidly (ϵ < 0.1) and consistently outperforms the usual FTSTM methodology even in the presence of considerable noise, as seen in Fig. 4c–j.
Application to real data
The results obtained above on synthetic data show that SBDSTM is able to recover kernels even at high defect density and in the presence of significant white noise, where alternate techniques such as manual detection fail. However, one might still question whether the method will work on real STM data, which might have other types of noise or errors, which we have not accounted for in our synthetic data. In order to investigate this fully, we now apply SBDSTM to investigate a set of experimentally obtained STM images of NaFe_{1−x}Co_{x}As at different values of x = 0.0, 0.015 and 0.02. At x = 0.0 (parent compound), no additional cobalt dopants are present in the lattice, and the only defects present are those intrinsic to the crystal, which are at a concentration of about 1%. This compound has been previously studied via STM and the data have been analyzed using the standard FT technique^{18}. To contrast with the standard FTSTM analysis, we implement SBDSTM on raw experimental data to demonstrate the significant improvement in data fidelity of the ReFT. Shown in Fig. 5b, c are the recovered Kernel and activation map, respectively, from the map shown in Fig. 5a. Figure 5d, e show the ReFT of the entire image and of the kernel, respectively, over the same range in Fourier space. The phase sensitive recovery of the FT in Fig. 5e when compared to Fig. 5d is immediately apparent. We note that for this particular doping the individual kernels are well separated, and the kernel can be isolated directly by eye from the large area image in Fig. 5a. We show the application of SBDSTM to this sample to illustrate that the recovered kernel is indeed what one would expect by direct measurement around an individual defect.
We now turn to the sample with x = 0.015, a STS image of which is shown in Fig. 5f. At this doping, we see that there are regions of highdensity clustering of the kernels, where the clustering is dense enough such that the individual kernels overlap and become difficult to resolve. In Fig. 5g, h, we show the corresponding recovery for the kernel and activation map, respectively. The ReFT of the entire map and the recovered kernel are shown in Fig. 5i, j, respectively. At this defect level, we can very occasionally see isolated defects, and the recovered kernel is seen to nicely match with the differential conductance around these isolated defects.
Finally, we consider a sample that is optimally doped (highest T_{c}) with a nominal cobalt concentration of x = 0.02. Shown in Fig. 5k is an STS image obtained on this sample at T = 5.9 K. At this doping level, there are no isolated impurities present anywhere in the sample, and the kernel cannot be manually recovered. The SBDSTM is able to recover the kernel and activation map as shown in Fig. 5l, m respectively. The ReFT of the entire image and of the recovered kernel are shown in Fig. 5n, o, respectively. We note that the kernel recovered at this doping level is much closer to fourfold symmetry than at lower cobalt dopings. This is consistent with previous measurements of the phase diagram of NaFe_{1−x}Co_{x}As^{36}, including those by STM^{37} that have shown that the transition from orthorhombic to tetragonal symmetry happens before optimal doping. We see from the series of data in Fig. 5 that SBDSTM works over the entire doping range that is relevant to STM experiments and is able to recover highquality kernels with phasesensitive FTs.
From the results shown on both synthetic and real experimental STM data, we can see that SBDSTM provides a complete recovery of kernels in real space and therefore a phase sensitive recovery of the FT in reciprocal space. Within the formulation of QPI, the phase of the FT in the QPI signal is dependent on the incoming and outgoing quasiparticle’s Green’s function as well as the potential of the impurity. The availability of phase information can give us new insight into individual materials that is not available simply from the magnitude of the QPI signal. In the remainder of this paper, we consider one such new insight into the physics of NaFe_{1−x}Co_{x}As. NaFe_{1−x}Co_{x}As, like many of the pnictides displays a superconducting dome as a function of cobalt doping. The maximum T_{c} at optimal doping (x = 0.02) reaches 18 K. As with the other pnictides, determining the symmetry of the superconducting order parameter in this compound is of much current interest. In this context, recent theoretical work on QPI in the superconducting state of the pnictides^{29,38,39} has opened up the possibility of distinguishing different superconducting order parameters from their QPI signature. We follow the procedure outlined in the recent work by Altenfeld et al. ^{39}, where the (real) Fourier transform of the QPI signal around a single defect δρ(q, ω) is integrated over all qspace to produce the quantity δρ(ω). This quantity is then antisymmetrized with respect to energy relative to the Fermi level to produce a quantity δρ^{−}(ω) = δρ(ω) − δρ(−ω). It is shown^{29,39} that in the case of s^{±} pairing, δρ^{−}(ω) is large and of constant sign over the energy range near the superconducting gap. Conversely, in the case of s^{++} pairing, δρ^{−}(ω) is expected to be small and have a sign change around the gap energy. In order to carry out the integration over q described in this procedure, it is required that we have access to the phase of the QPI signal. One way of achieving this is to directly image around an isolated dopant or defect where the complete phase sensitive pattern can be measured. Such STS imaging has recently been performed on iron chalcogenides^{40,41}. However, this method has not been applied to the iron arsenides, especially at optimal doping where the defect or dopant density is high and the phase information in the FT was not previously available. Armed with the phasesensitivity of SBDSTM, we now analyze differential conductance maps of near optimally doped NaFe_{1−x}Co_{x}As to investigate the superconducting order parameter at optimal doping.
We start with a dataset that consists of 21 raw STS images from −10 to +10 meV in 1 meV increments on an optimally doped NaFe_{1−x}Co_{x}As sample at T = 5.9 K. One of these raw images at ω = −10 meV is shown in Fig. 6a (additional images are described in Supplementary Note 10). Notice that no individual motifs can be resolved by eye. We then proceed to recover the kernels \(\hat{{\mathcal{A}}}({\bf{r}},\omega)\) at each energy using SBDSTM. An example of this recovery is shown in Fig. 6b, which is the recovered SBDSTM kernel \(\hat{{\mathcal{A}}}\)(r, ω = − 10 meV), recovered from the raw STS image in Fig. 6a. The recovered kernel lacks the strong anisotropy of recovered kernels from the underdoped regime, suggesting that electronic nematicity is not strong at this doping. Assuming that SBDSTM has worked correctly, the recovered \(\hat{{\mathcal{A}}}({\bf{r}},\omega)\) is identical to the realspace QPI signal δρ(r, ω). We then take the real part of the 2D Fourier transform \(\hat{{\mathcal{A}}}({\bf{q}},\omega)\), as shown in Fig. 6c at ω = −10 meV. This FT has the fullphase information present, and we can then integrate over q and antisymmetrize with respect to energy:
We perform this procedure at each energy, and plot the resultant \({\hat{{\mathcal{A}}}}^{}(\omega)\) in Fig. 6d. In Fig. 6e, we show the spatially averaged differential conductance from the same datasets as a function of energy, revealing the superconducting gap. From the two coherence peaks, we calculate a 2Δ of 11 meV. We can clearly see from Fig. 6d that \({\hat{{\mathcal{A}}}}^{}(\omega)\) is peaked near the superconducting gap, and has no sign change in the energy range near the gap. For comparison we performed theoretical calculations of the antisymmetrized correction to the LDoS, δρ^{−}(ω) as shown in the inset of Fig. 6d following the original prescription^{29}. Here, we use the electronic structure of Codoped NaFeAs previously measured by angleresolved photoemission spectroscopy (ARPES)^{42,43} and fitted to the 10 orbital tightbinding model^{44}. The values of the superconducting gap on each band were taken to be Δ_{h} = 6.5 meV, Δ_{e} = 6.8 meV on the electron and the smaller hole pockets, respectively^{42}, and Δ_{H} = 3.5 meV on the larger hole pocket^{43}. Further details are given in Supplementary Note 9. As expected, the behavior of the \({\hat{{\mathcal{A}}}}^{}(\omega)\) for the signchanging gaps between electron and hole pockets follows the predicted behavior of the δρ^{−}(ω) for an s^{+−}pairing state in which this quantity does not change in an energy range between the gaps on the hole and electron pockets, while for signpreserving gaps this quantity is generally small, with an alternating sign between the gaps. A similar behavior is found in the experiment, as shown in the main Fig. 6d. This procedure illustrates some of the new physical insight into STS image data that can be obtained once the complete phase information in the QPI signal is available for analysis.
Discussion
In its current implementation, SBDSTM addresses the problem of identifying a single motif across a series of images. Beyond the identification of realspace motifs in microscopy images, SBDSTM can also be applied to problems in which the motif is sparse in the appropriately chosen space, such as sparsity in the spatial gradient for natural image deblurring^{45,46}. Moreover, the flexibility of the convolutional data model in (3) affords the natural generalization, \({\mathcal{Y}}={\sum }_{j=1}^{M}{{\mathcal{A}}}_{0}^{\left(j\right)} \otimes {{\mathcal{X}}}_{0}^{\left(j\right)}+{\mathcal{Z}}\), which expands the scope of SBDSTM to identify multiple distinct kernels in any series of images. In particular, STM images that contain various shortrange orders, such as charge or spin density waves, would be amenable to a similar analysis^{47,48,49,50}. SBDSTM recovered results from these STM measurements can be directly compared with theoretical predictions^{51,52,53} to understand the nature of competing orders in superconductors and other strongly correlated materials. Other analysis methodologies^{47,50,54} have been recently proposed to improve FT data fidelity and provide some phasesensitive information on the structure of ordered phases. These alternative approaches provide compelling information under suitable conditions, but their results are still vulnerable to phase noise contamination. The correct implementation of SBDSTM to such cases remains an open but solvable problem.
Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
We thank Ethan Rosenthal and Erick Andrade for help with STM data acquisition, and Andrew Millis and Rafael Fernandes for discussions. M.A.M. and I.M.E. are thankful to Sergey Borisenko for providing the tightbinding parametrization of the ARPES data in NaFe_{1−x}Co_{x}As. This work is supported by the National Science Foundation Bigdata program (Grant number IIS1546411). Support for STM equipment and operations is provided by the Air Force Office of Scientific Research (Grant number FA95501610601). The work of I.M.E. was carried out with financial support from the Ministry of Science and Higher Education of the Russian Federation in the framework of Increase Competitiveness Program of NUST MISiS Grant No. K22017085.
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S.C.C. and J.Y.S. applied the SBD algorithm to STM data. Y.L., Z.C., J.S., and Y.Z. developed the SBD algorithm. M.A.M. and I.M.E. performed theoretical calculations of QPI in NaFeAs. J.N.W. and A.N.P. advised.
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Cheung, S.C., Shin, J.Y., Lau, Y. et al. Dictionary learning in Fouriertransform scanning tunneling spectroscopy. Nat Commun 11, 1081 (2020). https://doi.org/10.1038/s41467020146331
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DOI: https://doi.org/10.1038/s41467020146331
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